Self-similar fractals and self-similar energies

We discuss the existence and uniqueness of diffusions adapted to self-similar finitely ramified fractals. An adapted diffusion has a space time scaling with respect to the self-similar scaling of the fractal and visits every open subset of the fractal. Following Lindstrom the existence and uniqueness is reduced to a finite dimensional nonlinear eigenvalue problem for the renormalization map acting on all possible energies of the fractal graph. Simplified versions of recent stability results for the renormalization map and its reducibility properties in the case of non-unique diffusions are discussed.